ON THE ICOSAHEDRON INEQUALITY OF L´ASZL´O FEJES

COMPUTATIONS WITH MATHEMATICA 10, AN ADDENDUM TO THE PAPER
”ON THE ICOSAHEDRON INEQUALITY OF LÁSZLÓ FEJES-TÓTH”.
ÁKOS G.HORVÁTH
In this note we publish those symbolic and numerical calculations which need to the paper in title.
We used Mathematica 10 and its translating program into Latex.
In[1]: D[1/6Sin[c](Cos[(x-c)/2]-Cos[x/2]Cos[c/2])/(1-Cos[x/2]Cos[c/2]), x,c]
Out[1]:
−
1
x
Cos[ 2c ]Cos[c](−Cos[ 2c ]Cos[ x
2 ]+Cos[ 2 (−c+x)])Sin[ 2 ]
2
12(1−Cos[ 2c ]Cos[ x
2 ])
+
(−Cos[ 2c ]Cos[ x2 ]+Cos[ 12 (−c+x)])Sin[ 2c ]Sin[c]Sin[ x2 ]
+
2
24(1−Cos[ 2c ]Cos[ x
2 ])
1
1
1
c
x
1
c
x
1
1
Sin[c]( 4 Cos[ 2 (−c+x)]− 4 Sin[ 2 ]Sin[ 2 ])
Cos[c]( 2 Cos[ 2 ]Sin[ 2 ]− 2 Sin[ 2 (−c+x)])
+
−
6(1−Cos[ 2c ]Cos[ x
6(1−Cos[ 2c ]Cos[ x
2 ])
2 ])
1
c
1
c
x
1
c
x
1
x
Cos[ x
Sin
Sin[c]
Cos
Sin
−
Sin
(−c+x)
Cos
Sin[c]Sin
Cos
[2]
(2 [2] [2] 2 [2
])
[2]
[ 2 ]( 2 [ 2 ]Sin[ 2c ]+ 12 Sin[ 12 (−c+x)])
2]
−
2
2
12(1−Cos[ 2c ]Cos[ x
12(1−Cos[ 2c ]Cos[ x
2 ])
2 ])
c
x
1
c
x
Cos[ 2c ]Cos[ x
2 ](−Cos[ 2 ]Cos[ 2 ]+Cos[ 2 (−c+x)])Sin[ 2 ]Sin[c]Sin[ 2 ]
3
12(1−Cos[ 2c ]Cos[ x
2 ])
+
In[2]: Simplify[%]
((
[ ]
[x]
[ c ]3
Out[2]:
9Cos 3c
Cos[x]+
2 − 2(5 + 10Cos[c] + Cos[2c])Cos 2 + Cos 2
( (
[
]
)
[
])
[
]
[ x ])3 )
1
c
c
c
Cos
(30
+
(11
+
3Cos[c])Cos[x])
Sin
/
96
−1
+
Cos
Cos
2
2
2
2
2
In[3]: D[1/6Sin[c](Cos[(x-c)/2]-Cos[x/2]Cos[c/2])/(1-Cos[x/2]Cos[c/2]), {x,2}]
[ ]
[ ]
[
])
( 41 Cos[ 2c ]Cos[ x2 ]− 14 Cos[ 12 (−c+x)])Sin[c] 1 (
Out[3]:
+ 6 −Cos 2c Cos x2 + Cos 12 (−c + x) ·
6(1−Cos[ 2c ]Cos[ x
])
2
(
)
2
2
1
c
x
1
1
Cos[ 2c ]Cos[ x
Cos[ 2c ] Sin[ x
Cos[ 2c ]Sin[c]Sin[ x
2]
2]
2 ]( 2 Cos[ 2 ]Sin[ 2 ]− 2 Sin[ 2 (−c+x)])
Sin[c] −
−
2 +
3
2
c
x
c
x
4(1−Cos[ 2c ]Cos[ x
2
1−Cos
Cos
6
1−Cos
Cos
])
(
[
]
[
])
(
[
]
[
])
2
2
2
2
2
In[4]: Simplify[%]
Out[4]:
c
Sin[ 2c ]Sin[c](−2Cos[c]Sin[ x
2 ]+Cos[ 2 ]Sin[x])
3
48(−1+Cos[ 2c ]Cos[ x
2 ])
In[5]: D[1/6Sin[c](Cos[(x-c)/2]-Cos[x/2]Cos[c/2])/(1-Cos[x/2]Cos[c/2]), {c,2}]
(
)
2
c 2
(
[c]
[x]
[1
])
Cos[ 2c ]Cos[ x
Cos[ x
1
2]
2 ] Sin[ 2 ]
Out[5]:
−
Sin[c] +
2 +
3
6 −Cos 2 Cos 2 + Cos 2 (−c + x)
4(1−Cos[ 2c ]Cos[ x
2(1−Cos[ 2c ]Cos[ x
2 ])
2 ])
((
[
]
[
]
[
])
c
x
1
1
1
1
4 Cos 2 Cos 2 − 4 Cos 2 (−c + x) Sin[c]−
6(1−Cos[ 2c ]Cos[ x
2 ])
(
[ ]
[ ]
[
])
(
[ ]
[ ]
[
]))
− −Cos 2c Cos x2 + Cos 21 (−c + x) Sin[c] + 2Cos[c] 12 Cos x2 Sin 2c + 21 Sin 12 (−c + x)
−
[
]
[
]
(
(
[
]
[
]
x
c
1
Cos[c] −Cos 2c Cos x2 +
2 Cos 2 Sin 2
6(1−Cos[ 2c ]Cos[ x
2 ])
[
])
(
[ ]
[
]))
[ ]
Cos 12 (−c + x) + Sin[c] 21 Cos x2 Sin 2c + 12 Sin 12 (−c + x)
In[6]: Simplify[%]
[
]
(
]
[
]
[
− 11Sin[2c − x]+
Out[6]:
4Sin 12 (c − 3x) + Sin 21 (5c − 3x) − 24Sin[c − x] + 3Sin 3(c−x)
2
[
]
[1
]
[1
]
39Sin 2 (3c − x) + Sin 2 (5c − x) − 6Sin[x] + 24Sin[c + x] − 3Sin 3(c+x)
+ 11Sin[2c + x]−
2
[1
]
[1
]
[1
]
[1
])
− 39Sin 2 (3c + x) − Sin 2 (5c + x) − 4Sin 2 (c + 3x) − Sin 2 (5c + 3x) / (384 (−1+
[ ]
[ ])3 )
+Cos 2c Cos x2
Date: 2014 Nov.
1
2
Á. G.HORVÁTH
In[7]: D[1/6Sin[c](Cos[(x − c)/2] − Cos[x/2]Cos[c/2])/(1 − Cos[x/2]Cos[c/2]), {x, 2}] ·
· D[1/6Sin[c](Cos[(x − c)/2] − Cos[x/2]Cos[c/2])/(1 − Cos[x/2]Cos[c/2]), {c, 2}] −
D[1/6Sin[c](Cos[(x − c)/2] − Cos[x/2]Cos[c/2])/(1 − Cos[x/2]Cos[c/2]), x, c]∧ 2
(
(
)
2
c 2
])
Cos[ 2c ]Cos[ x
Cos[ x
2]
2 ] Sin[ 2 ]
Out[7]:
−Cos 2 Cos 2 + Cos 2 (−c + x)
−
Sin[c]+
2 +
3
4(1−Cos[ 2c ]Cos[ x
2(1−Cos[ 2c ]Cos[ x
2 ])
2 ])
((
[
]
[
]
[
])
(
[
]
[
]
[
])
1
1
c
x
1
1
c
x
1
Cos
−
Sin[c]
−
−Cos
Cos
+
Cos
Sin[c]+
Cos
Cos
(−c
+
x)
(−c
+
x)
c
x
4
2
2
4
2
2
2
2
6(1−Cos[ 2 ]Cos[ 2 ])
(1
[x]
[c] 1
[1
])) (
[x]
[c](
(
[c]
[x]
[1
])
2Cos[c] 2 Cos 2 Sin 2 + 2 Sin 2 (−c + x)
− Cos 2 Sin 2 Cos[c] −Cos 2 Cos 2 + Cos 2 (−c + x) +
(
(
[ ]
[ ]
[
]))) ( (
[ ]
[ ])2 )) ( 41 Cos[ 2c ]Cos[ x2 ]− 14 Cos[ 12 (−c+x)])Sin[c]
+
Sin[c] 21 Cos x2 Sin 2c + 12 Sin 12 (−c + x)
/ 6 1 − Cos 2c Cos x2
6(1−Cos[ 2c ]Cos[ x
2 ])
(
)
2
2
(
[c]
[x]
[1
])
Cos[ 2c ]Cos[ x
Cos[ 2c ] Sin[ x
1
2]
2]
−Cos
Cos
+
Cos
(−c
+
x)
Sin[c]
−
+
−
2
3
x
x
c
c
6
2
2
2
4 1−Cos[ 2 ]Cos[ 2 ])
2(1−Cos[ 2 ]Cos[ 2 ])
) ( (
1
1
c
x
1
1
x
Cos[ 2c ]Sin[c]Sin[ x
Cos[ 2c ]Cos[c](−Cos[ 2c ]Cos[ x
2 ]( 2 Cos[ 2 ]Sin[ 2 ]− 2 Sin[ 2 (−c+x)])
2 ]+Cos[ 2 (−c+x)])Sin[ 2 ]
− −
+
2
2
c
x
6(1−Cos[ 2c ]Cos[ x
12
1−Cos
Cos
(
[ 2 ] [ 2 ])
2 ])
c
x
1
c
x
Cos[ 2c ]Cos[ x
(−Cos[ 2c ]Cos[ x2 ]+Cos[ 12 (−c+x)])Sin[ 2c ]Sin[c]Sin[ x2 ]
2 ](−Cos[ 2 ]Cos[ 2 ]+Cos[ 2 (−c+x)])Sin[ 2 ]Sin[c]Sin[ 2 ]
+
+
3
2
12(1−Cos[ 2c ]Cos[ x
24(1−Cos[ 2c ]Cos[ x
])
2
2 ])
1
1
1
c
x
1
c
x
1
1
Sin[c]( 4 Cos[ 2 (−c+x)]− 4 Sin[ 2 ]Sin[ 2 ])
Cos[c]( 2 Cos[ 2 ]Sin[ 2 ]− 2 Sin[ 2 (−c+x)])
+
−
6(1−Cos[ 2c ]Cos[ x
6(1−Cos[ 2c ]Cos[ x
2 ])
2 ])
)2
c
1
c
x
1
1
1
x
c
1
1
Cos[ x
Cos[ 2c ]Sin[c]Sin[ x
2 ]Sin[ 2 ]Sin[c]( 2 Cos[ 2 ]Sin[ 2 ]− 2 Sin[ 2 (−c+x)])
2 ]( 2 Cos[ 2 ]Sin[ 2 ]+ 2 Sin[ 2 (−c+x)])
−
2
2
12(1−Cos[ 2c ]Cos[ x
12(1−Cos[ 2c ]Cos[ x
2 ])
2 ])
1
6
(
[c]
[x]
[1
In[8]: Simplify[%]
((
[
]
[
]
Out[8]:
336 + 502Cos[c] + 256Cos[2c] + 26Cos[3c] − 66Cos 21 (c − 3x) − 14Cos 21 (5c − 3x) −
[
]
[
]
[
]
3(c−x)
2Cos 21 (7c − 3x) + 7Cos[c − 2x] + Cos[3c − 2x] − 498Cos c−x
+
190Cos[c
−
x]
−
46Cos
+
2
2
[1
]
[1
]
4Cos[2(c − x)] + 100Cos[2c − x] − 298Cos 2 (3c − x) + 18Cos[3c − x] − 98Cos 2 (5c − x) −
[
]
[
]
[
]
3(c+x)
2Cos 12 (7c − x) + 280Cos[x] + 8Cos[2x] − 498Cos c+x
+
190Cos[c
+
x]
−
46Cos
+
2
2
[1
]
[1
]
4Cos[2(c + x)] + 100Cos[2c + x] − 298Cos 2 (3c + x) + 18Cos[3c + x] − 98Cos 2 (5c + x) −
]
[
]
[
]
[
2Cos 12 (7c + x) + 7Cos[c + 2x] + Cos[3c + 2x] − 66Cos 12 (c + 3x) − 14Cos 12 (5c + 3x) −
[
])
[ ]2 ) (
(
[ ]
[ ])5 )
2Cos 12 (7c + 3x) Sin 2c
/ 576 −2 + 2Cos 2c Cos x2
In[9]: Plot3D[1/6Sin[c](Cos[(x-c)/2]-Cos[x/2]Cos[c/2])/(1-Cos[x/2]Cos[c/2]), {x,0,Pi/2}, {c,x,2ArcSin[Sqrt[2/3]]}]
Out[9]:
3
In[10]: D[1/6Sin[c](Cos[(x-c)/2]-Cos[x/2]Cos[c/2])/(1-Cos[x/2]Cos[c/2]), c]
Out[10]:
+
Sin[c](
1
Cos[c](−Cos[ 2c ]Cos[ x
2 ]+Cos[ 2 (−c+x)])
6(1−Cos[
[ ]Sin[ ]
]
1
x
2 Cos 2
c
2
c
2
6(1−Cos[
[
[ ])
c
2
]Cos[ ])
])
x
2
−
c
x
1
c
Cos[ x
2 ](−Cos[ 2 ]Cos[ 2 ]+Cos[ 2 (−c+x)])Sin[ 2 ]Sin[c]
12(1−Cos[ 2c ]Cos[ x
2 ])
+ 12 Sin 12 (−c+x)
Cos x
2
In[11]: Simplify[%]
Out[11]:
(
)
c 3
Sin[ 2c ] (1+3Cos[c])Sin[ x
2 ]−2Cos[ 2 ] Sin[x]
2
12(−1+Cos[ 2c ]Cos[ x
2 ])
In[12]: Plot3D[Out[11], {x,0,Pi/2}, {c,x,2ArcSin[Sqrt[2/3]]}]
Out[12]:
In[13]: RegionPlot[Out[11]>=0 ,{x,0,Pi/2},{c,0,2ArcSin[Sqrt[2/3]]}]
Out[13]:
2
+
4
Á. G.HORVÁTH
In[14]: Plot3D[Out[8], {x,0,Pi/2}, {c,x,2ArcSin[Sqrt[2/3]]}]
Out[14]:
In[15]:RegionPlot[Out[8]>=0 ,{x,0,Pi/2},{c,0,2ArcSin[Sqrt[2/3]]}]
Out[15]:
In[16]: Reduce[ Out[8]==0 && x==Pi/2 && 0 < c < 2ArcSin[Sqrt[2/3]], {x,c}]
]
[√
[
]
2
3
4
π
Out[16] x == 2 &&c == 4ArcTan
Root 1 − 24#1 + 78#1 − 24#1 + #1 &, 1
In[17]: N[%]
Out[17]: x == 1.5708&&c == 0.875793
In[18]: Solve[z ∧ 4 − 24z ∧ 3 + 78z ∧ 2 − 24z + 1==0]
{{
√
√
√
√ } {
√
√ }
Out[18]:
z → 6 − 34 − 35 − 6 34 , z → 6 − 34 + 35 − 6 34 ,
{
√
√
√
√ } {
√
√ }}
z → 6 + 34 − 35 + 6 34 , z → 6 + 34 + 35 + 6 34
5
In[19]: N[%]
Out[19]:
{{z → 0.049513}, {z → 0.288583}, {z → 3.46521}, {z → 20.1967}}
In[20]: Reduce[ Out[8]==0 && c==Pi/2 && 0 < x < Pi/2, {x,c}]
[√
]
√
√
2 5
3 2√
√
Out[20]: x == 4ArcTan
− 10+7
+
&&c == π2
2
10+7 2
In[21] N[%]
Out[21]:
x == 0.427922&&c == 1.5708
In[22]: Reduce[ Out[8]==0 && c==2ArcSin[Sqrt[2/3]] && 0¡x¡Pi/2, {x,c}]
[
[√ ]]
[√
[{
[
[√ ]]
√
2
2
+
10368Cos
4ArcSin
+
Out[22]: x == 4ArcTan Root 14712 − 256 3 + 20331Cos 2ArcSin
3
3
[
[√ ]] (
[
[√
]]
[
[√
]]
√
2
2
2
+1053Cos 6ArcSin
+ 51168 − 3072 3 + 81324Cos 2ArcSin
+ 41472Cos 4ArcSin
+
3
3
3
[
[√ ]])
(
[
[√ ]]
[
[√ ]]
2
2
2
+ 4212Cos 6ArcSin
#1 + 66768 + 121986Cos 2ArcSin
+ 62208Cos 4ArcSin
+
3
3
3
[
[√ ]])
(
[
[√
]]
√
2
2
+ 6318Cos 6ArcSin
#12 + + 51168 + 3072 3 + 81324Cos 2ArcSin
+
3
3
[
[√ ]]
[
[√ ]])
(
[
[√ ]]
√
2
2
2
+41472Cos 4ArcSin
+ 4212Cos 6ArcSin
#13 + 14712 + 256 3 + 20331Cos 2ArcSin
+
3
3
3
[
[√ ]]
[
[√ ]])
}]]
[√ ]
2
2
2
+10368Cos 4ArcSin
+ +1053Cos 6ArcSin
#14 &, 0.03105
&&c == 2ArcSin
3
3
3
In[23]: N[%]
Out[23]: x == 0.697715&&c == 1.91063
In[24]:Reduce[ Out[8]==0 && x==-2Pi/3+2ArcSin[Sqrt[14]/4] && x¡c¡2ArcSin[Sqrt[2/3]], {x,c}]
(
[ √ ])
7
√
√
√
[√
[{
2
Out[24]: x == − 3 π − 3ArcSin 2 2
&&c == 4ArcTan Root 2699 − 768 2 + 163 21 − 448 42+
[ (
[ √ ])]
[ (
[ √ ])]
7
7
2
4
2
+560Cos 3 π − 3ArcSin 2
+ 16Cos 3 π − 3ArcSin 2 2
+
[ √ ])]
(
[ (
7
√
√
√
2
+
+ −18227 + 4992 2 − 2059 21 + 2592 42 + 3920Cos 3 π − 3ArcSin 2 2
[ (
[ √ ])])
7
√
√
(
+112Cos 43 π − 3ArcSin 2 2
#1 + 16167 − 9600 2 + 1695 21−
[ √ ])]
[ (
[ √ ])])
[ (
7
7
√
2
4
2
+ 336Cos 3 π − 3ArcSin 2 2
#12 +
−3360 42 + 11760Cos 3 π − 3ArcSin 2
(
[ (
[ √ ])]
7
√
√
√
2
+ −2751 + 9216 2 − 3447 21 − 2304 42 + 19600Cos 3 π − 3ArcSin 2 2
+
[ (
[ √ ])])
7
√
√
√
(
+560Cos 43 π − 3ArcSin 2 2
#13 + −2751 − 9216 2 − 3447 21 + 2304 42+
[ (
[ √ ])]
[ (
[ √ ])])
7
7
2
4
2
+19600Cos 3 π − 3ArcSin 2
+ 560Cos 3 π − 3ArcSin 2 2
#14 +
[ √ ])]
(
[ (
7
√
√
√
2
+
16167 + 9600 2 + 1695 21 + 3360 42 + 11760Cos 3 π − 3ArcSin 2 2
[ √ ])])
[ (
7
√
√
√
(
#15 + −18227 − 4992 2 − 2059 21 − 2592 42+
+336Cos 43 π − 3ArcSin 2 2
[ √ ])]
[ (
[ √ ])])
[ (
7
7
√
(
4
2
2
+ 112Cos 3 π − 3ArcSin 2 2
#16 + 2699 + 768 2+
+3920Cos 3 π − 3ArcSin 2
[ √ ])]
[ (
7
√
√
+
+163 21 + 448 42 + 560Cos 23 π − 3ArcSin 2 2
6
Á. G.HORVÁTH
[ (
[ √ ])])
}]]
7
7
4
2
+16Cos 3 π − 3ArcSin 2
#1 &, 0.1324047495312671
In[25]: N[%]
Out[25]: x == 0.324463&&c == 1.39593
In[26]: Reduce[ Out[8]==0 && x==-4ArcSin[Sqrt[14]/4]+5Pi/3 && x¡c¡Pi/2, {x,c}]
(
[ √ ])
7
√
√
√
[√
[{
Out[26]: x == 13 5π − 12ArcSin 2 2
&&c == 4ArcTan Root 9361 − 5376 3 − 2432 7 + 933 21+
[ (
[ √ ])]
[ (
[ √ ])]
7
7
√
√
(
+2240Cos 13 5π − 12ArcSin 2 2
+64Cos 23 5π − 12ArcSin 2 2
+ −55113 + 31104 3 + 15168 7−
[ (
[ √ ])]
[ (
[ √ ])])
7
7
√
−11709 21 + 15680Cos 13 5π − 12ArcSin 2 2
+ 448Cos 23 5π − 12ArcSin 2 2
#1+
(
[ (
[ √ ])]
7
√
√
√
+ 51813 − 40320 3 − 25920 7 + 9225 21 + 47040Cos 13 5π − 12ArcSin 2 2
+
[ (
[ √ ])])
7
√
√
√
(
+1344Cos 23 5π − 12ArcSin 2 2
#12 + 17811 − 27648 3 + 13824 7 − 19377 21+
[ (
[ √ ])]
[ (
[ √ ])])
7
7
+78400Cos 13 5π − 12ArcSin 2 2
+ 2240Cos 23 5π − 12ArcSin 2 2
#13 +
(
[ (
[ √ ])]
7
√
√
√
+ 17811 + 27648 3 − 13824 7 − 19377 21 + 78400Cos 13 5π − 12ArcSin 2 2
+
[ (
[ √ ])])
7
√
√
√
(
+2240Cos 23 5π − 12ArcSin 2 2
#14 + 51813 + 40320 3 + 25920 7 + 9225 21+
[ (
[ √ ])]
[ (
[ √ ])])
7
7
+47040Cos 13 5π − 12ArcSin 2 2
+ 1344Cos 23 5π − 12ArcSin 2 2
#15 +
(
[ (
[ √ ])]
7
√
√
√
+ −55113 − 31104 3 − 15168 7 − 11709 21 + 15680Cos 13 5π − 12ArcSin 2 2
+
[ (
[ √ ])])
7
+448Cos 23 5π − 12ArcSin 2 2
#16 +
(
[ (
[ √ ])]
7
√
√
√
+ 9361 + 5376 3 + 2432 7 + 933 21 + 2240Cos 13 5π − 12ArcSin 2 2
+
[ (
[ √ ])])
}]]
7
+64Cos 23 5π − 12ArcSin 2 2
#17 &, 0.1604829573029123
In[27]: N[%]
Out[27]: x == 0.398271&&c == 1.52411
In[28]:Reduce[Out[8]==0&&x==Pi/5&&0 < c < 2ArcSin[Sqrt[2/3]], {x, c}]
(
((
[{
x = π5 ∧ c = 4 tan−1 Root #12 − 5&, 2#1 + #22 − 10&, −2#1 + #32 − 10&, 29#1#47 −
189#1#46 +393#1#45 −105#1#44 −105#1#43 +393#1#42 −189#1#4+29#1+8#2#47 −60#2#46 +
156#2#45 − 288#2#44 + 288#2#43 − 156#2#42 + 60#2#4 − 8#2 + 56#3#47 − #3#46 +
420#3#45 + 288#3#44 − 288#3#43 − 420#3#42 + 324#3#4 − 56#3
+ 167#47 − 951#46 + 1083#45 +
}
])1/2 )
4
3
2
∨c=4
341#4 + 341#4 + 1083#4 − 951#4 + 167& , {2, 2, 2, 4}
((
[{ 2
2
2
−1
tan
Root #1 − 5&, 2#1 + #2 − 10&, −2#1 + #3 − 10&, 29#1#47 −
189#1#46 + 393#1#45 − 105#1#44 − 105#1#43 + 393#1#42 − 189#1#4 + 29#1 +
8#2#47 − 60#2#46 + 156#2#45 − 288#2#44 + 288#2#43 − 156#2#42 + 60#2#4 − 8#2 + 56#3#47 −
324#3#46 + 420#3#45 + 288#3#44 − 288#3#43 − 420#3#42 + 324#3#4 − 56#3 + ))
167#47 −
}
])
1/2
951#46 + 1083#45 + 341#44 + 341#43 + 1083#42 − 951#4 + 167& , {2, 2, 2, 5}
In[29]:N [%]
7
Out[29]: x = 0.628319 ∧ (c = 0.36077 ∨ c = 1.83487)
[{
sin(u)
sin(v)
sin(x)
sin(y)
sin(z)
In[30]: NMaximize √3−cos(u)
+ √3−cos(v)
+ √3−cos(x)
+ √3−cos(y)
+ √3−cos(z)
,
}
]
π
u + v + x + y + z ≤ 2 , 0 < x, 0 < y, 0 < z, 0 < u, 0 < v , {x, y, z, u, v}
Out[30]: {1.97836, {x → 0.314159, y → 0.314159, z → 0.314159, u → 0.314159, v → 0.314159}}